Special Cases

Several cases simplify Equation 4.38. If k 2 21 0, no copolymer is formed, 

This resembles the vapor y-liquid x composition at constant relative volatility a/. 

Terpolymers (three monomers) require six relative reactivities and can be handled conveniently only in special cases or by computer methods. 

The reactivity of a combination of monomers depends a great deal on the nature of the polymerizing center. For example, the reactivity ratios of butadiene (1) and styrene (2) are quite different in an anionic copolymerization (r, =5, Tj = 0.04) and in a radical copolymerization [r- = 1.39, Tj = 0.78). If a copolymerization is performed with a 50/50 mixture of the two, what are the initial compositions of the polymers formed in the two cases  

Solution Using Equation 4.38, the equation that provides the instantaneous copolymer composition, f, for the anionic polymerization case is 

6 Special Cases Special methods are required for quantitative extraction of some lipid classes such as gangliosides and polyphosphoinositides, and to minimize artifactual formation of NEFA, DAG, and polyphosphoinositides when 

Recently, a monophase strategy by using chloroform-methanol-water was practiced for broader lipid analysis including gangliosides [49]. Analysis of lipids present in a single rat retina is achieved in the study. 

It is useful to list certain special results for the case of viscous flow and molecular diffusion. Results for turbulent flow profiles and diffusion coefficients can be obtained by numerical integration as has been done by Taylor (1954) and more recently by Tichacek and others (1958). For molecular diffusion ipj = 1, and for the problem of diffusion only in the absence of a second phase, R = 13 = y = 1 giving a single factor k = - 2kx2 + i3. 

The results for diffusion only and for coated tube chromatography are contained in (21) and (22) of the preceding paragraph. We give here the results for diffusion and chromatography when the gas phase flows through the annulus. If the liquid phase is stationary (22) may be used in conjunction with these. 

As tr- 0 these reduce to the values implicit in (21) for the open tube. As a— 1 

Thus for the parallel walled channel the Taylor diffusion coefficient is 

DISPERSION IN COAXIAL STREAMS OF ARBITRARY CROSS-SECTION 

There are certain special cases that warrant developing a specific program to exploit their peculiarities to the fullest is opportune. There are two situations in which this 

flash separators, distillation columns, and reactor networks might all be good examples of systems that warrant special programs to improve their solutions. 

There are special values of the parameter x for which Eqs.(37) are not valid. For cases of integer or half-integer x = j, which correspond to the special cases of the model (35), in Eq. (27) one can easily recognize Maleev s boson representation of spin S = j operators [28]  

Generally, the wave functions (4) and (26) resemble the MP form but with infinite matrices represented by the Bose operators. However, in accordance to Maleev s representation in the special cases the infinite matrices formed by the Bose operators b+ and b can be broken off to the size n = 2j + 1, and wave function (26) is reduced to the usual MP form 

Exact calculation of the correlators in the thermodynamic limit is performed using standard transfer matrix technique and results in 

In the particular case of zero singlet weight, y — 0, when spins on each rung form a local triplet, correlation functions (42) coincide with those obtained in [9, 29], 

According to Eqs.(42) the spin correlations have an exponential decay and the correlation length rc is 

For analysis purposes, the equations for conservation mass and the first law of thermodynamics are simplified with some assumptions. Two such common cases are (i) the steady-state steady-flow process and (ii) the uniform-flow uniform-state process. 

Flow and energy transfer across a control volume. 

A considerable number of examples will be found in the text in which halo-genated and other substituted olefins are produced. Their modes of fonnation do not usually differ in principle from the corresponding unsubstituted case. However, some special methods have been used, for example, the direct preparation of halo olefins from ketones with phosphorous hahdes, or via hydrazones. 

Not all leads are discovered through a high-throughput screen of a chemical library. Two of the more commonly mentioned alternative methods include serendipitous discovery and optimization of side effects from an existing drug or drug candidate. The important role of natural products in drug discovery also deserves extra attention. 

From Equation (2.69) we have = and therefore E = Q In this case, 

When yr = nll, the light wave propagates in the yz plane. Eigenmode 1 is = (1,0,0), where T stands for the transpose, which is a long thex axis, and the corresponding eigenvalue is 

For polarizer 2, the transmission axis is parallel to the y axis in the lab frame. In the local frame x y z , the transmission axis is parallel to the y axis and the uniaxial axis is parallel to the z axis. For the eigenmode that has the non-absorbing refractive index rig and can pass polarizer 2, the direction of the eigen electric field vector is along the direction which in the local frame is given by 

The position of the double or triple bond is indicated by the number of the lowest carbon atom in the alkene or alkyne. 

2-methylbuta-1,3-diene (an example of a conjugated diene two C=C bonds separated by one C-C bond) 

For atkenes, the position of the C=C bond is sometimes shown at the front of the name, e.g. 2-pentene 

Monosubstituted benzene derivatives are usually named after benzene (CgHg), although some non-systematic or common names (in brackets) are still used. 

Reactions of benzene and substituted benzenes are diseussed in Chapter 7 

There are several groups of persons who require an additional risk assessment due to their being more at risk than other groups. Three such groups will be considered - young persons, expectant and nursing mothers and disabled workers. 

More detailed guidance is available from HSE Books -HSG165 Young people at work and HSG 199 Managing health and safety on work experience. 

Typical factors which might affect such women are  

Detailed guidance is available in New and expectant mothers at work, HSG122, HSE Books. 

Organizations have been encouraged for many years to employ workers with disabilities and to ensure that their premises provide suitable access for such people. From a health and safety point of view, it is important that workers with a disability are covered by special risk assessments so that appropriate controls are in place to protect them. For example, employees with a hearing problem will need to be warned when the fire alarm sounds or a fork lift truck approaches. Special vibrating signals or flashing lights may be used. Similarly workers in wheelchairs will require a clear, wheelchair friendly, route to a fire exit and onwards to the assembly point. Safe systems of work and welfare facilities need to be suitable for any workers with disabilities. 

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For the special case of a bubble-point calculation (incipient vaporization), a is 0 (also Q = 0) and Equation (7-13) becomes... 

Separation of mixtures of condensable and non-condensable components. If a fluid mixture contains both condensable and noncondensable components, then a partial condensation followed by a simple phase separator often can give a food separation. This is essentially a single-stage distillation operation. It is a special case that deserves attention in some detail later. 

In general, the final network design should be achieved in the minimum number of units to keep down the capital cost (although this is not the only consideration to keep down the capital cost). To minimize the number of imits in Eq. (7.1), L should be zero and C should be a maximum. Assuming L to be zero in the final design is a reasonable assumption. However, what should be assumed about C Consider the network in Fig. 7.16, which has two components. For there to be two components, the heat duties for streams A and B must exactly balance the duties for streams E and F. Also, the heat duties for streams C and D must exactly balance the duties for streams G and H. Such balemces are likely to be unusual and not easy to predict. The safest assumption for C thus appears to be that there will be one component only, i.e., C = 1. This leads to an important special case when the network has a single component and is loop-free. In this case, ... 

This type of analysis requires several chromatographic columns and detectors. Hydrocarbons are measured with the aid of a flame ionization detector FID, while the other gases are analyzed using a katharometer. A large number of combinations of columns is possible considering the commutations between columns and, potentially, backflushing of the carrier gas. As an example, the hydrocarbons can be separated by a column packed with silicone or alumina while O2, N2 and CO will require a molecular sieve column. H2S is a special case because this gas is fixed irreversibly on a number of chromatographic supports. Its separation can be achieved on certain kinds of supports such as Porapak which are styrene-divinylbenzene copolymers. This type of phase is also used to analyze CO2 and water. 

For a long time the official specifications for diesel fuel set only a mciximum viscosity of 9.5 mm /s at 20°C. Henceforth, a range of 2.5 mm /s minimum to 4.5 mm /s maximum has been set no longer for 20°C but at 40°C which seems to be more representative of injection pump operation. Except for special cases such as very low temperature very fluid diesel fuel and very heavy products, meeting the viscosity standards is not a major problem in refining. 

The volume fraction of water (S J and the saturation exponent n can be considered as expressing the increased difficulty experienced by an electrical current passing through a partially oil filled sample. (Note is only a special case of C, when a reservoir... 

In the special case when a = 0, any symmetric 2-D tensor field can be represented as... 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

The measuren.cnts are performed on a sample made from Inconell IN 718 placed at our disposal by Me oren- und Turbinen-Union Miinchen GmbH (MTU). The sample is of disk shape and has a curved side and a flat side, see also fig. 6. On the curved side MTU had detected a segregation reaching to the surface but with unknown depth. It had been detected by segregation etching and its existence was confirmed by eddy current testing. Both methods are successful only in such a special case where the segregation extends to the surface. 

Equation 11-3 is a special case of a more general relationship that is the basic equation of capillarity and was given in 1805 by Young [1] and by Laplace [2]. In general, it is necessary to invoke two radii of curvature to describe a curved surface these are equal for a sphere, but not necessarily otherwise. A small section of an arbitrarily curved surface is shown in Fig. II-3. The two radii of curvature, R and / 2[Pg.6]

A monolayer can be regarded as a special case in which the potential is a square well however, the potential well may take other forms. Of particular interest now is the case of multilayer adsorption, and a reasonable assumption is that the principal interaction between the solid and the adsorbate is of the dispersion type, so that for a plane solid surface the potential should decrease with the inverse cube of the distance (see Section VI-3A). To avoid having an infinite potential at the surface, the potential function may be written... 

A special case of adsorption in cavities is that of clatherate compounds. Here, cages are present, but without access windows, so for adsorption to occur the solid usually must be crystallized in the presence of the adsorbate. Thus quinol crystallizes in such a manner that holes several angstroms in diameter occur and, if crystallization takes place in the presence of solvent or gas... 

If a surface reaction is bimolecular in species A and B, the assumption is that the rate is proportional to 5a x 5b- We now proceed to apply this interpretation to a few special cases. 

Only two of the many possible special cases need be considered. Thus if the products and reactants are weakly adsorbed,... 

One serious issue is the detemihiation of the exchange energy per particle, e, or the corresponding exchange potential, V The exact expression for either of these quantities is miknown, save for special cases. If one... 

Rigid linear molecules are a special case in which an extended MS group, rather than the MS group, is isomorphic to the point group of the equilibrium structure see chapter 17 of [1]. 

In order to specify fhe size of fhe sysfem, af leasf one of fhese variables ought to be extensive (one that is proportional to the size of the system, like n or the total volume V). In the special case of several phases in equilibrium several extensive properties, e.g. n and Vfor two phases, may be required to detennine the relative amounts of the two phases. The rest of the variables can be intensive (independent of the size of the system) like T,p, the molar volume V = V/n,or the density p. For multicomponent systems, additional variables, e.g. several ns, are needed to specify composifion. 

Equation (A2.1.21) includes, as a special case, the statement dS > 0 for adiabatic processes (for which Dq = 0) and, a fortiori, the same statement about processes that may occur in an isolated system (Dq = T)w = 0). If the universe is an isolated system (an assumption that, however plausible, is not yet subject to experimental verification), the first and second laws lead to the famous statement of Clausius The energy of the universe is constant the entropy of the universe tends always toward a maximum. ... 

The special case of equation (A2.1.27) when T and p are constant (dJ= 0, dp = 0) is called the Gibbs-Duhem equation, so equation (A2.1.27) is sometimes called the generalized Gibbs-Duhem equation . 

Note that in this special case, the heat absorbed directly measures a state fiinction. One still has to consider how this constant-volume heat is measured, perhaps by an electric heater , but then is this not really work Conventionally, however, if work is restricted to pressure-volume work, any remaining contribution to the energy transfers can be called heat . 

Although the exact equations of state are known only in special cases, there are several usefid approximations collectively described as mean-field theories. The most widely known is van der Waals equation [2]... 

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

Ss ) Pp are simultaneously zero for this special case, this point is at x = 1/2 and... 

The final scattering angle 0 is defined rising 0 = 0(t = oo). There will be a correspondence between b and 0 that will tend to look like what is shown in figure A3.11.5 for a repulsive potential (liere given for the special case of a hard sphere potential). 

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